Problem: Factor the following expression: $-7$ $x^2$ $-27$ $x+$ $40$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(40)} &=& -280 \\ {a} + {b} &=& & & {-27} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-280$ and add them together. Remember, since $-280$ is negative, one of the factors must be negative. The factors that add up to ${-27}$ will be your ${a}$ and ${b}$ When ${a}$ is ${8}$ and ${b}$ is ${-35}$ $ \begin{eqnarray} {ab} &=& ({8})({-35}) &=& -280 \\ {a} + {b} &=& {8} + {-35} &=& -27 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-7}x^2 +{8}x {-35}x +{40} $ Group the terms so that there is a common factor in each group: $ ({-7}x^2 +{8}x) + ({-35}x +{40}) $ Factor out the common factors: $ x(-7x + 8) + 5(-7x + 8) $ Notice how $(-7x + 8)$ has become a common factor. Factor this out to find the answer. $(-7x + 8)(x + 5)$